A possible drawback of the ordinary correlation coefficient $ρ$ for two real random variables $X$ and $Y$ is that zero correlation does not imply independence. In this paper we introduce a new correlation coefficient $ρ^*$ which assumes values between zero and one, equalling zero iff the two variables are independent and equalling one iff the two variables are linearly related. The coefficients $ρ^*$ and $ρ^2$ are shown to be closely related algebraically, and they coincide for distributions on a $2\times 2$ contingency table. We derive an orthogonal decomposition of $ρ^*$ as a positively weighted sum of squared ordinary correlations between certain marginal eigenfunctions. Estimation of $ρ^*$ and its component correlations and their asymptotic distributions are discussed, and we develop visual tools for assessing the nature of a possible association in a bivariate data set. The paper includes consideration of grade (rank) versions of $ρ^*$ as well as the use of $ρ^*$ for contingency table analysis. As a special case a new generalization of the Cram{é}r-von Mises test to $K$ ordered samples is obtained.